Maximum performance in self-compatible thermoelectric elements
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V. Pluschke Institute of Mathematics, University Halle-Wittenberg, D-06099 Halle, Germany
C. Goupil Laboratoire CRISMAT, UMR 6508, Caen, France
K. Zabrocki and E. Müller Institute of Materials Research, German Aerospace Center (DLR), D-51170 Köln, Germany
G.J. Snyder California Institute of Technology, Pasadena, California 91125 (Received 14 December 2011; accepted 11 April 2011)
Within the framework of a new optimization strategy based on self-compatible thermoelectric elements, the ability to reach maximum performance is discussed. For the efficiency of a thermogenerator and the coefficient of performance of a Peltier cooler, the constraint z T 5 ko 5 const. turned out to provide a suitable criterion for judging maximum performance. In this paper ko is calculated as an average of the temperature dependent figure of merit z T.
I. INTRODUCTION
The efficiency g of a thermogenerator (TEG) is commonly defined as the net electrical power output, – P, divided by the thermal power Qh supplied, whereas the coefficient of performance (COP) [ u of a Peltier cooler (TEC) is defined as the ratio of the cold side cooling power (Qc) by the electrical power (P) supplied: P g¼ Qh
and
Qc u¼ P
:
ð1Þ
Global maximization of these performance parameters is a suitable guideline for an empirical approach to thermoelectric (TE) device optimization. Nevertheless, it is more attractive for optimization purposes to express both g and u in terms of the intensive variables as are relative current density u and reduced “efficiencies” gr and ur that are used as primary values within the framework of the compatibility approach1; for an overview see also Refs. 2–5. In this paper, we use the notation for a unified model of TEG/TEC under steady-state conditions introduced in Refs. 5 and 6, where the absorbing side denoted with index “a” in a generator is the hot side (index “h”) and in a cooler is the cold side (index “c”). For the sink side denoted with the index “s” it is vice versa. In particular, we refer to a prismatic TE element of length L, cross-sectional area Ac, and fixed boundary temperatures Ta and Ts (see Fig. 1), where Ta is the temperature at the heat absorbing side
(Ta 5 Th for TEG, but Ta 5 Tc for TEC), and Ts denotes the heat sink temperature that is often fixed to the room temperature. Assuming parallelity of heat flux and electrical current density (flow direction x axis), we end up in a one-dimensional description, where the relative current density is defined as u¼
j j T 0 ðxÞ
;
ð2Þ
with j 5 I/Ac being the electrical current density. II. REDUCED EFFICIENCIES AND COMPATIBILITY FACTORS FOR TEG AND TEC
Considering the COP of a TEC first, we find within the framework of the constant properties model u¼
a Ta j j ð@T=@xÞjx¼0 a ðTs Ta Þ j þ q L j2
;
ð3Þ
with Seebeck coefficient a, resistivity q 5 1/r, isothermal electrical conductivity r, and thermal conductivity j (under zero current). Taking the limits for an infinitesimal segment (L ! dx; DT ¼ Ts Ta ! dT, etc.), we obtain 0 1 a T j j T 0 ðxÞ T @a T j j T 0 ðxÞA : ¼
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